Lagrange

Lagrange

Lagrange, Joseph Louis, Comte, 1736-1813, French mathematician and astronomer, b. Turin, of French and Italian descent. Before the age of 20 he was professor of geometry at the royal artillery school at Turin. With his pupils he organized (1759) a society from which the Turin Academy of Sciences developed. Among his early successes were his method of solving isoperimetrical problems, on which the calculus of variations is based in part; his researches on the nature and propagation of sound and on the vibration of strings; and his studies on the libration of the moon and on the satellites of Jupiter. On the recommendation of Euler and D'Alembert, Frederick the Great invited him (1766) to succeed Euler as director of mathematics at the Berlin Academy of Sciences. The memoirs of the academy were enriched by his distinguished treatises, and during this time he wrote his chief work, Mécanique analytique, a treatment of mechanics based solely on algebra and the calculus and containing not a single diagram or geometric explanation. This was published (1788) in Paris, where he had been called by Louis XVI in 1787. In 1793 he became president of the commission on weights and measures; he was influential in causing the adoption of the decimal base for the metric system. A professor at the École polytechnique from 1797, he developed the use in teaching of the analytic method that he so skillfully employed in his research. He wrote Théorie des fonctions analytiques (1797) and Leçons sur le calcul des fonctions (1806), both based on his lectures. Under Napoleon, Lagrange was made senator and count; he is buried in the Panthéon. His contributions to the development of mathematics also include the application of differential calculus to the theory of probabilities and notable work on the solution of equations. In astronomy he is known for his calculations of the motions of planets.

Proof of Lagrange's Theorem

This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. If we can show that all cosets of H have the same number of elements, then we are done, since H itself is a coset of H. Now, if aH and bH are two left cosets of H, we can define a map f : aH → bH by setting f(x) = ba-1x. This map is bijective because its inverse is given by f −1(y) = ab−1y.

This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G : H] (the number of left cosets of H in G). If we write this statement as

|G| = [G : H] · |H|,

then, interpreted as a statement about cardinal numbers, it remains true even for infinite groups G and H.

Using the theorem

A consequence of the theorem is that the order of any elementa of a finite group (i.e. the smallest positive integer k with ak = e) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows

Existence of subgroups of a certain order

Lagrange's theorem raises the question of whether every divisor of the order of a group is the order of a subgroup. This need not hold. Given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d. The smallest example is the alternating groupG = A4 which has 12 elements but no subgroup of order 6. Any finite group which has a subgroup with
size equal to any (positive) divisor of the size of the group must be solvable, so nonsolvable groups are examples of this phenonenon, although A4 shows that they aren't the only examples.
If G is abelian, then there always exists a subgroup of any order dividing the size of G. A partial generalization is given by Cauchy's theorem.

History

Lagrange did not prove Lagrange's theorem in its general form.
What he actually proved was that if a polynomial in n variables has its variables permuted in all n! ways, the number of
different polynomials that are obtained is always a factor of n!. (For example if the variables x, y, and z are permuted in all 6 possible ways in the
polynomial x + y - z then we get a total of 3 different polynomials: x + y − z, x + z - y, and y + z − x. Note 3 is a factor of 6.)
The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations which
preserve the polynomial. (For the example of x + y − z, the subgroup H in S3 contains the identity and the transposition (xy).)
So the size of H divides n!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to
the general theorem about finite groups which now bears his name.